diff --git a/rsa.py b/rsa.py
index 788a85c1c8a7c1251b8b2a47200f56aebc4402d8..df7401d360b78655ff0a2dd2c6658a8c8130b8c8 100644
--- a/rsa.py
+++ b/rsa.py
@@ -1,20 +1,21 @@
"""
-Project : TP numéro 2 : RSA
-Lesson : IT 121 – 2021-2022
-Authors : Florian BURGENER, Gawen ACKERMANN, Quentin FASLER, Dario GENGA
+Project : Travail Pratique RSA
+Course : ISC_121 – 2021-2022
+Authors : Dario GENGA, Florian BURGENER, Gawen ACKERMANN, Quentin FASLER,
Date : 2021-2022
"""
+import math
def get_bezout_coefficients(a, b):
- """Find the Bézout coefficients of a and b.
+ """Find the Bézout coefficients for the numbers a and b.
Args:
a (int): The number a
b (int): The number b.
Returns:
- tuple: Bézout coefficients.
+ tuple: The Bézout coefficients.
"""
r = [a, b]
x = [1, 0]
@@ -24,24 +25,26 @@ def get_bezout_coefficients(a, b):
while r[i] > 0:
i += 1
+
r.append(r[i - 2] % r[i - 1])
q.append(int(r[i - 2] / r[i - 1]))
if r[i] > 0:
x.append(x[i - 2] - q[i] * x[i - 1])
y.append(y[i - 2] - q[i] * y[i - 1])
+
return x[-1], y[-1]
def modular_inverse(a, n):
- """Compute the modular inverse of a number a modulo n.
+ """Calculates the modular inverse of a number a modulo n.
Args:
- a (int): The number to reverse.
+ a (int): The number to be reversed.
n (int): The modulo.
Returns:
- int: The reversed number.
+ int: The inverted number.
"""
coefficients = get_bezout_coefficients(a, n)
@@ -50,119 +53,72 @@ def modular_inverse(a, n):
return None
-def get_q_p(n):
- """Get the value of q and p from n by bruteforce.
+def break_encryption(n):
+ """Breaks RSA encryption using the brute force technique.
Args:
- n (int): One of the two value of the public key.
+ n (int): The component of the public key which is the product of p and q.
Returns:
- tuple: The two value that represent n.
+ tuple: The prime numbers p and q.
"""
- p = 2
- while True:
- if n % p == 0:
- q = n // p
- return (p, q)
- p += 1
+ range_low = 3
+ range_high = int(math.ceil(math.sqrt(n)))
+
+ for x in [2] + list(range(range_low, range_high, 2)):
+ if n % x == 0:
+ return (x, n // x)
+
+ return None
def modular_pow(base, exponent, modulus):
- """Compute the modular power.
+ """Computes the modular exponentiation.
Args:
- base (int): The value of a part of the message to be decoded.
- exponent (int): The private key.
- modulus (int): One of the two value of the public key.
+ base (int): Power base.
+ exponent (int): Power exponent.
+ modulus (int): The modulos.
Returns:
- int: The int value of the value, ready to be decoded to utf-8.
+ int: The result of the exponentiation.
"""
if modulus == 1:
return 0
+
result = 1
base %= modulus
+
while exponent > 0:
if exponent % 2 == 1:
result = (result * base) % modulus
+
exponent = exponent >> 1
base = (base * base) % modulus
+
return result
def main():
e = 5249
n = 1653973759
- messages = (
- 1511395078,
- 260436590,
- 1630654276,
- 1190458520,
- 790492067,
- 515550941,
- 297140366,
- 755589582,
- 647075331,
- 1191707844,
- 901889430,
- 660956124,
- 1500654109,
- 984322720,
- 1275630738,
- 1244853107,
- 1445928913,
- 1312523810,
- 265093060,
- 933013993,
- 1375592761,
- 195866064,
- 534502441,
- 928270408,
- 166404031,
- 621272622,
- 1304987439,
- 905393335,
- 55120151,
- 772595721,
- 506609577,
- 1172751778,
- 162439707,
- 233959833,
- 1468937795,
- 1358701120,
- 901889430,
- 495995733,
- 1524090698,
- 1043509086,
- 934992314,
- 1545639379,
- 1061595897,
- 1348452679,
- 1135067876,
- 905393335,
- 621272622,
- 55120151,
- 233959833,
- 1220119699,
- 708711266,
- 517797467,
- 195866064,
- 1579814353,
- 412378626,
- 498875436,
- 445485200,
- 7656659
- )
-
- p, q = get_q_p(n)
+ encrypted_data = (1511395078, 260436590, 1630654276, 1190458520, 790492067, 515550941, 297140366, 755589582, 647075331, 1191707844, 901889430, 660956124, 1500654109, 984322720, 1275630738, 1244853107, 1445928913, 1312523810, 265093060, 933013993, 1375592761, 195866064, 534502441, 928270408, 166404031, 621272622, 1304987439, 905393335, 55120151, 772595721, 506609577, 1172751778, 162439707, 233959833, 1468937795, 1358701120, 901889430, 495995733, 1524090698, 1043509086, 934992314, 1545639379, 1061595897, 1348452679, 1135067876, 905393335, 621272622, 55120151, 233959833, 1220119699, 708711266, 517797467, 195866064, 1579814353, 412378626, 498875436, 445485200, 7656659)
+
+ p, q = break_encryption(n)
+
+ # Calculation of the decryption key.
d = modular_inverse(e, (p - 1) * (q - 1))
- result = ""
- for x in messages:
- tmp = modular_pow(x, d, n)
- mBytes = tmp.to_bytes(tmp.bit_length(), byteorder="little")
- result += mBytes.decode("utf-8")
+ decrypted_data = []
+
+ for x in encrypted_data:
+ decrypted_data.append(modular_pow(x, d, n))
+
+ decoded_data = ""
+
+ for x in decrypted_data:
+ decoded_data += x.to_bytes((x.bit_length() + 7) // 8, "little").decode("utf-8")
- print(result)
+ print(decoded_data)
if __name__ == "__main__":
diff --git a/annexe /Group_13_Enonce.txt "b/\303\251nonc\303\251/RSA_Groupe_13.txt"
similarity index 100%
rename from annexe /Group_13_Enonce.txt
rename to "\303\251nonc\303\251/RSA_Groupe_13.txt"
diff --git a/annexe /enonce_tp_RSA.pdf "b/\303\251nonc\303\251/TP2-RSA.pdf"
similarity index 100%
rename from annexe /enonce_tp_RSA.pdf
rename to "\303\251nonc\303\251/TP2-RSA.pdf"